A120871 - OEIS

%I #20 Jul 11 2020 03:04:47

%S 1,4,2,7,8,17,7,18,17,32,14,31,9,28,23,46,16,41,34,63,25,56,14,47,36,

%T 73,23,62,49,92,34,79,64,113,47,98,28,81,62,119,41,100,79,142,56,121,

%U 31,98,73,144,46,119,92,169,63,142,113,196,82,167,49,136,103,194

%N a(n) is the value of j for row n of the fixed-j dispersion for Q = 8.

%C This sequence results from A087056 by deleting duplicates.

%H Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling2/kimberling45.html">The equation (j+k+1)^2 - 4*k = Q*n^2 and related dispersions</a>, Journal of Integer Sequences, 10 (2007), Article #07.2.7.

%e For each positive integer n, there is a unique pair (j,k) of positive integers such that (j + k + 1)^2 - 4*k = 8*n^2. This representation is used to define the fixed-j dispersion for Q = 8, given by A120860, having northwest corner:

%e   1,  5,  29, 169, ...

%e   2, 10,  58, 338, ...

%e   3, 17,  99, 577, ...

%e   4, 22, 128, 746, ...

%e   ...

%e The pair (j,k) for each n, shown in the position occupied by n in the above array, is shown here:

%e   (1,2), (1,14),  (1,82),  (1,478), ...

%e   (4,1), (4,25), (4,161),  (4,953), ...

%e   (2,7), (2,47), (2,279), (2,1631), ...

%e   (7,4), (7,56), (7,356), (7,2104), ...

%e   ...

%e The fixed-j for row 1 is a(1) = 1;

%e the fixed-j for row 2 is a(2) = 4; etc.

%e (For example, (4 + 25 + 1)^2 - 4*25 = 8*10^2.)

%o (PARI) f(n) = 3*n + 2*sqrtint(2*n^2); \\ A098021

%o unused(listus) = {my(v=vecsort(Vec(listus))); for (i=1, vecmax(v), if (!vecsearch(v, i), return (i)); ); };

%o D(nb) = {my(m = matrix(nb, nb), t); my(listus = List); for (g=1, nb, if (g==1, t = 1, t = unused(listus)); m[g, 1]=t; listput(listus, t); t = f(t); m[g, 2]=t; listput(listus, t); for (h=3, nb, t = 6*m[g, h-1] - m[g, h-2]; m[g, h] = t; listput(listus, t); ); ); m; }; \\ A120860

%o q(n) = 2*n^2 - sqrtint(2*n^2)^2; \\ A087056

%o lista(nn) = my(m=D(nn)); vector(nn, n, q(m[n, 1])); \\ _Michel Marcus_, Jul 09 2020

%Y Cf. A087056, A098021, A120860.

%K nonn

%O 1,2

%A _Clark Kimberling_, Jul 10 2006

%E More terms from _Michel Marcus_, Jul 09 2020